A normalization formula for the Jack polynomials in superspace and an identity on partitions

TL;DR

This paper proves a conjectured formula for the norm of Jack polynomials in superspace, using combinatorial methods and identities on partitions, advancing understanding of symmetric functions in superspace.

Contribution

It provides a closed-form formula for the norm of Jack polynomials in superspace, confirmed through combinatorial proofs and identities on partitions.

Findings

Established a closed-form norm formula for Jack polynomials in superspace.

Developed combinatorial proof techniques involving admissible tableaux.

Derived an identity on weighted sums of partitions using Gessel-Viennot methods.

Abstract

We prove a previously conjectured closed form formula for the norm of the Jack polynomials in superspace with respect to a certain scalar product. The proof is mainly combinatorial and relies on the explicit expression in terms of admissible tableaux of the non-symmetric Jack polynomials. In the final step of the proof appears an identity on weighted sums of partitions that we demonstrate using the methods of Gessel-Viennot.